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On the convergence of $$\sum c_kf(n_kx)$$
István Berkes, Graz University of Technology, Austria, and Michel Weber, Université Louis-Pasteur et C.N.R.S., Strasbourg, France
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Memoirs of the American Mathematical Society
2009; 72 pp; softcover
Volume: 201
ISBN-10: 0-8218-4324-9
ISBN-13: 978-0-8218-4324-6
List Price: US$62 Individual Members: US$37.20
Institutional Members: US\$49.60
Order Code: MEMO/201/943

Let $$f$$ be a periodic measurable function and $$(n_k)$$ an increasing sequence of positive integers. The authors study conditions under which the series $$\sum_{k=1}^\infty c_k f(n_kx)$$ converges in mean and for almost every $$x$$. There is a wide classical literature on this problem going back to the 30's, but the results for general $$f$$ are much less complete than in the trigonometric case $$f(x)=\sin x$$. As it turns out, the convergence properties of $$\sum_{k=1}^\infty c_k f(n_kx)$$ in the general case are determined by a delicate interplay between the coefficient sequence $$(c_k)$$, the analytic properties of $$f$$ and the growth speed and number-theoretic properties of $$(n_k)$$. In this paper the authors give a general study of this convergence problem, prove several new results and improve a number of old results in the field. They also study the case when the $$n_k$$ are random and investigate the discrepancy the sequence $$\{n_kx\}$$ mod 1.

• Discrepancy of random sequences $$\{S_n x\}$$